This is especially relevant for constructing quantum field theory on curved space-time (QFTCS). Canonical approaches to quantum theory require a Hilbert space on which to represent the canonical commutation relations, field equations, operator algebra, etc. In QFTCS, the problem is that there are an infinite number of such representations, each of which are unitarily inequivalent, and with no way to choose among them.As a general rule, the issue of unitarily inequivalent Hilbert space representations has to do with the infinite dimensional configuration space of field theories; in such cases, the Stone-von Neumann theorem (see e.g., [1]} for a reference), which holds for finite-dimensional configuration spaces, fails. Consequently, issues can also crop up in flat space-time; for instance, with gauge theories [2]}. In recent years, the development of QFTCS has thus favored approaches that do not commit to a particular Hilbert space representation. A popular example of this is the local algebraic quantum field theory (AQFT) approach initiated by Haag [3]}, and more recently pursued by, for example, by Wald and Hollands (see e.g., [4]}[5]}).
